4-Connected Triangulations on Few Lines GD 2019 September 20., - - PowerPoint PPT Presentation

4-Connected Triangulations

• n Few Lines

GD 2019 September 20., 2019 Pr˚ uhonice/Prague Stefan Felsner (TUB, Berlin)

Line Cover Number

π(G) = min

• ℓ : ∃plane drawing of G with vertices covered by ℓ lines
• Classes with π(G) = 2:
• trees
• outerplanar
• grids

Lower bound

Theorem [ Eppstein, SoCG 19 ]. ∃ planar, bipartite, cubic, 3-connected graphs Gn with π(Gn) ∈ Ω(n1/3).

Lower bound

• Corollary. ∃ planar 4-connected graphs Gn with π(Gn) ∈ Ω(n1/3).

Our contribution

• Theorem. For all G planar 4-connected π(G) ≤

√ 2n. Tools:

• planar lattices
• orthogonal partitions of posets
• transversal structures

Posets & Lattices

3 diagrams.

• a planar poset of dimension 3
• a non-planar lattice
• a planar lattice.

Theorem. A finite lattice is planar if and only if its dimension is ≤ 2.

A planar lattice and its conjugate

1 2 3 4 5 6 7 8 9 1011121314 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 1011121314

Chains, antichains, width, and height

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 1011121314 1 2 3 4 5 6 7 8 9 1011121314 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Canonical chain and antichain partitions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 1011121314 1 2 3 4 5 6 7 8 9 1011121314 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Planar lattices on h lines

• Proposition. A planar lattice of height h has a diagram with

points on h horizontal lines.

Greene–Kleitman theory

With a poset P with n elements there is a partition λ of n, such that for the Ferrer’s diagram G(P) of λ we have:

• The number of squares in the ℓ longest columns of G(P)

equals the maximal number of elements covered by an ℓ-chain.

• The number of squares in the k longest rows of G(P) equals

the maximal number of elements covered by a k-antichain.

maximal 2-chain maximal 3-antichain

Orthogonal pairs

A chain family C and an antichain family A are orthogonal iff 1. P =

A∈A

A

• ∪

C∈C

C

• , and

2. |A ∩ C| = 1 for all A ∈ A, C ∈ C. Theorem [ Frank ’80 ]. If (ℓ, k) is a corner of G(P), then there is an orthogonal pair consisting of a ℓ-chain C and a k-antichain A.

Orthogonal pairs

Theorem [ Frank ’80 ]. If (ℓ, k) is on the boundary of G(P), then there is an orthogonal pair consisting of a ℓ-chain C and a k-antichain A.

• Corollary. A poset with n elements has (ℓ, k) an orthogonal pair

consisting of a ℓ-chain C and a k-antichain with k + ℓ ≤ √ 2n − 1.

Planar lattices on ≤ √ 2n − 1 lines

• Proposition. A planar lattice with n elements has a diagram with

points on k horizontal lines and ℓ vertical lines where k + ℓ ≤ √ 2n − 1.

Adjusting chains and antichains

• Lemma. C, A an orthogonal pair of P
• C′ the canonical chain partition of PC
• A′ the canonical antichain partition of PA

= ⇒ C′, A′ is an orthogonal pair of P.

Canonical chain cover

• Lemma. (C1, . . . , Cℓ) the canonical chain partition of PC

= ⇒ there are extensions C +

i

• f Ci such that
• C +

i

is a maximal chain of PC

• C +

i

⊆

j≤i Cj

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

• draw Ci and add left ears to Ci
• add the connecting edges, chains, components

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

C +

i−1

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

• draw Ci and add left ears to Ci

Ci

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

• draw Ci and add left ears to Ci

Ci

Phase i

In phase i we add all elements between C +

i−1 and C + i

including C +

i

• add right ears to C +

i−1

• draw Ci and add left ears to Ci
• add the connecting edges, chains, components

Transversal structures

• Proposition. 4-connected inner triangulations of a quadrangle

admit transversal structures (a.k.a. regular edge labeling). b a t s b t a s

Transversal structures and planar lattices

• Proposition. The red graph of a transversal structure is the

diagram of a planar lattice. t s

4-connected planar

• Proposition. The blue edges can be included while drawing the

red lattice.

4-connected planar

1 extra line for the missing edge.

Thank you